论文标题
硬件感知和稳定的正交框架
A Hardware-aware and Stable Orthogonalization Framework
论文作者
论文摘要
正交过程是Krylov空间方法中必不可少的基础，它占用了计算时间的很大一部分。常用的方法，例如革兰氏 - schmidt方法，分别考虑投影和归一化，并显式存储正交基础。我们将正交化和归一化的问题视为QR分解问题，在该问题上，我们应用了已知算法，即Choleskyqr和TSQR。这导致方法可以通过降低通信成本来解决正交问题，同时保持稳定性并将正交基础存储在局部正交表示中。此外，我们讨论了一种新方法作为一种框架，该框架使我们能够结合不同的正交算法，并为硬件的每个部分使用最佳算法。在制定方法之后，我们根据绩效模型来显示其有利的性能属性，该绩效模型将数据传输在计算节点内以及在计算节点之间的消息传递。理论结果通过数值实验验证。
The orthogonalization process is an essential building block in Krylov space methods, which takes up a large portion of the computational time. Commonly used methods, like the Gram-Schmidt method, consider the projection and normalization separately and store the orthogonal base explicitly. We consider the problem of orthogonalization and normalization as a QR decomposition problem on which we apply known algorithms, namely CholeskyQR and TSQR. This leads to methods that solve the orthogonlization problem with reduced communication costs, while maintaining stability and stores the orthogonal base in a locally orthogonal representation. Furthermore, we discuss the novel method as a framework which allows us to combine different orthogonalization algorithms and use the best algorithm for each part of the hardware. After the formulation of the methods, we show their advantageous performance properties based on a performance model that takes data transfers within compute nodes as well as message passing between compute nodes into account. The theoretic results are validated by numerical experiments.
